The core conjecture for qualitative independence graphs

A graph is a core if every endomorphism is an automorphism. Let $QI(k^2,k)$ be the qualitative independence graph.

Core conjecture. For any positive integer $k$, the graph $QI(k^2,k)$ is a core.

The case $k=3$, namely $QI(9,3)$, is known by work cited in the paper. The conjecture asks for the analogous rigidity for all $k$.